|HP-16C Encryption Algorithm|
Message #1 Posted by John Williamson on 18 Jan 2006, 7:56 p.m.
Here's a slightly unusual HP-16C program -- it's an encryption program based around the shrinking linear feedback register (lfsr) stream cipher (see Wikipedia's article of the SLFSR). The program takes plaintext blocks and a key and encodes (or decodes) a message. This algorithm is particularly suited to the 16c, where variable length registers are easy to deal with, and a bit count instruction is available (the feedback of an lfsr can be computed at bitcount(state&tap)&1).
Running the encryption on the calculator makes it very difficult for a potential adversary to tamper with the device or the program to weaken this system -- it's also very portable (and deniable, for the truly paranoid). Unfortunately, this algorithm is slow, it takes about a minute to encrypt a 64 bit block.
The cipher key should specify the lengths of 2 lfsrs, which should be between about 30 and 64 (max) bits each. The two lsfrs should be different lengths (the periods -- which are [2^(len) - 1] -- should be mutually prime for maximum effectiveness). Each length should be stored, as a 64 bit word, in register C and D respectively. The key should also specify the feedback polynomial to be used. Some example polynomials are given in the table at the end of this message. The two polynomial codes should be stored in register 2 and 9 respectively, with WSIZE set to the width of the appropriate lfsr. The key should finally give the initial state of the lfsr's, and again with WSIZE set to the width of the lfsr, these should be placed in registers 1 and 8.
To ensure security, you should generate two sequences of random digits (one for each lfsr) for each new message transmitted, and xor them with the original lfsr state as specified with the key. These digits must be noted and prepended [unencrypted] to the start of the transmitted message. This ensures that each run will not use identical keys.
Encryption proceeds by storing the first 64 bits of plaintext in register E, and executing GTO 0 then R/S. The output will be stored back in register E. The next 64 bits should be stored in E, and the algorithm repeated until the entire message has been completed. Decryption is identical, using the encoded messages as the plaintext.
Here's an example usage session which should make things clearer (maybe):
Given a key
len1 len2 poly 1 poly 2 state 1 state2
0x21 0x29 0x1b7536ba 0x1d56e7fe0fd 0xbb72d45d 0xb1bb10ff
imagine we want to encrypt the message HELLO WORLD. Rewriting this as alphabet indices, 31=space, we have
8 5 12 12 15 31 23 15 18 12 4
We could code this as 5 bit blocks and concatenate (which is more efficient), but I'll just use 8-bit
blocks here, so we have two 64 bit message blocks:
08 05 0c 0c 0f 1f 17 0f
18 0c 04 00 00 00 00 00
OK, to set up the encryption with this key, execute
(1) HEX/40/WSIZE 21 STO C (sets length of lfsr1)
(2) 29 STO D (sets length of lfsr2)
Now, the system is set up and we can encrypt messages! So do
40/WSIZE/08050c0c011f170f/STO E (this is the first part of the message we created above) GTO 0/R/S. After some time, we get the encrypted version back. Note this down. Then do 180c040000000000/STO E/GTO 0/R/S to encrypt the next part (and so on, if we had more blocks to do).
(3) 21/WSIZE/1b7536ba STO 2 (sets tap poly. for lfsr 1)
(4) 10bfea/ENTER (random digits just made up for this message, note these for later)
(5) bb72d45d/XOR/STO 1 (sets state for lfsr 1)
(6) 29/WSIZE/1d56e7fe0fd STO 9 (sets tap poly. for lfsr 2)
(7) a10786/ENTER (again, randomly chosen for this message, take a note of them for transmission)
(8) b1bb10ff/XOR/STO 8 (sets state for lfsr 2)
The message to transmit is then just the random digits we made up in (7) and (8), plus the outputs after each run. Decryption is identical, just use the initial digits from the start of the message in steps (7) and (8) instead of creating new ones, and put the encoded message in E and run it...
This is currently thought to be reasonably secure (though there are some existing attacks, none are currently practical, needing large amounts of ciphertext (>4gb) with known poly. tap sequences). This
cipher will probably not stop a dedicated government agency, but it will stop most others, for now, and it's the only one I could fit on the 16c! Any comments or suggestions for optimization are most welcome.
; Shrinking LFSR encryption
; storage register use:
; 1 -- lfsr1 state (n1 bits)
; 2 -- lfsr1 taps (n1 bits)
; 8 -- lfsr2 state (n2 bits)
; 9 -- lfsr2 taps (n2 bits)
; b -- counter variable (64 bits)
; c -- length of lfsr 1 (64 bits) [n1]
; d -- length of lfsr 2 (64 bits) [n2]
; e -- plaintext (64 bits)
; note: strange things can happen with registers overlapping
; with different wsizes. if lfsr1 is much longer than lfsr2, it's
; possible lfsr1 will overlap with lfsr2 and break everything. Don't
; use lfsr of wildly different widths and you should be fine.
encrypt:lbl 0 ; entry point
wsize ; ensure 64 bit mode
hex ; hex mode
cf 4 ; clear carry
0 ; 0x40 = 64 decimal
sto b ; store loop count
sto i ; lfsr1 starts at reg 1
rcl c ; get length for lfsr1
gsb 2 ; gsb [lfsr] execute lfsr starting w/register 1. result in carry
rlc ; shift carry into 1's position
sto f ; store lfsr1 output bit into B (from carry)
sto i ; lfsr2 starts are register 8
rcl d ; get length for lfsr2
gsb 2 ; gsb [lfsr] execute lfsr starting w/register 4. result in carry
f? 4 ; check the carry state (lfsr output)
gsb 3 ; gsb [wrt] if it was one, use this bit to encrypt
f? 4 ; check carry state (1 if bit was used)
x!=0 ; if carry, test b>0, else jump unconditionally
gto 1 ; goto encloop
rtn ; finished!
;; compute an lfsr
lfsr: lbl 2
wsize ; set register len
rcl (i) ; get lfsr state from i
isz ; i++
rcl (i) ; tap from i
b# ; compute popcount(tap&lfsr) & 1
; (only last bit will be shifted into carry
; implied &1)
rrc ; rotate the result into the carry
dsz ; i--
rcl (i) ; get lfsr state from i
rrc ; rotate (output) carry onto end
sto (i) ; store state back
wsize ; set size back to 64 bits
;; Write -- encrypts one bit of plaintext
wrt: lbl 3 ; encrypt one bit of the plaintext
rcl e ; recall the plaintext
rcl f ; the output bit from first lfsr
xor ; xor the first bit
rr ; rotate round (after 64 rotates, each bit will have been xor'd once)
sto e ; store back again
sto b ; decrement loop counter
sf 4 ; set carry
;;;;;;;;;;;;;;;; end of listing
Here's a short table of feedback polynomials:
I have a much larger set of tables at http://www.dcs.gla.ac.uk/~jhw/poly-hex.zip.